“For me it remains an open question whether [this work]
pertains to the realm of mathematics or to that of art.”

- M.C. Escher

Contents

Page 1
Introduction
The Art of Alhambra
Our Area of Focus
Our Aim

Page 2
The Principals behind Tessellations
- Translation
- Rotation
- Reflection
- Glide Reflection

Page 3
Mathematics in Escher's Art
- Translation
- Rotation
- Glide Reflection
- Combination

Page 4
Our Original Tessellations
- The Catch (Translation)
- Under the Sea (Rotation)
- The Herd (Glide Reflection)
- Dumbo & Butterfly (Combination)


Page 5
Possible links with Architecture
Conclusion
References

Introduction

Maurits Cornelis Escher (or simply M.C. Escher) was born in Leeuwarden, Holland in 1898. He was never a good student during his childhood days. The whole of his school days was a nightmare, the only saving grace was his weekly 2 hour art lessons. Although he was considered by his art teacher for having a more than average talent for art, Escher scored a lowly seven. His art was not highly thought of by the examiners. Little did they know that Escher would eventually become a famous artist for creating higly imaginative artwork that marries the world of art and mathematics.

In his later years, he recalled his childhood days and wrote:

At high school in Arnhem, I was extremely poor at arithmetic and algebra because I had, and still have, great difficulty with the abstractions of numbers and letters. When, later, in stereometry [solid geometry], an appeal was made to my imagination, it went a bit better, but in school I never excelled in that subject. But our path through life can take strange turns

Despite having no formal training in mathematics, Escher created artwork that followed certain mathematical principles. His works included exploration of the 3 dimensional world, perspective, abstract mathematical solids, approaches to infinity and also the focus of this paper – the art of creating tessellations.

Tessellations are arrangements of closed shapes that completely cover the plan without overlapping and leaving gaps.

Escher was fascinated by all types of tessellations, regular and irregular, and he himself questioned the domain in which tessellations fell under: mathematics or art?

He wrote:
In mathematical quarters, the regular division of the plane has been considered theoretically . . . Does this mean that it is an exclusively mathematical question? In my opinion, it does not. [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature they are more interested in the way in which the gate is opened than in the garden lying behind it."

Mathematics has shown that there are only three regular shapes that can be used for a tiling. Escher exploited these basic patterns in his tessellations, and applied principles of translations, glide reflections and rotations to obtain a wide variety of patterns. The effects of his tessellations are usually both astounding and beautiful.

The Art Of The Alhambra

Escher’s fascination with tessellations, or periodic drawing division began when he briefly visited Alhambra, Spain in 1926. He was greatly inspired and tried to emulate a rhythmic theme on a plane surface himself. However, he was frustrated by his attempts to do so, as he could only produce some ugly, rigid four legged beasts which walked upside down on his drawing paper. It was only during the second visit in 1937 that he began a more serious study into the art of creating tessellations.

He was fascinated by the rich possibilities latent in the rhythmic division of a plane surface found in Moorish tessellations. He and his wife studied these artworks deeply and Escher finally came up with a complete practical system that he applied in his later artworks of metamorphosis and cycle prints.


Escher's Sketch of the tessellations in Alhambra, Spain

Our Area of Focus

As Escher's works cover many areas from perspective distortions to metamorphoses, as such we felt that to talk a little about everything would have been explaining too little of too many things. As such, we have singled in onto one particular type of his work, tessellations.

In addition, the team felt that it we wanted to keep our explanations simple and clear, as such the extensive use of Flash animation is was chosen as the medium to relay our thoughts.

Aim

We hope to achieve through this paper:

-a deeper understanding and appreciation of Escher’s tessellations

-to uncover the underlying mathematical principles behind his artwork

-based on principles that we have learned, we will also attempt to create some original tessellations of our own

-lastly, to explore the possibilities of his artwork in practical usage in areas of architecture such as space, form and façade treatment.